MATLAB. Advanced Mathematics and Mechanics Applications Using. Third Edition. David Halpern University of Alabama CHAPMAN & HALL/CRC

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1 Advanced Mathematics and Mechanics Applications Using MATLAB Third Edition Howard B. Wilson University of Alabama Louis H. Turcotte Rose-Hulman Institute of Technology David Halpern University of Alabama CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

2 IX Contents 1 Introduction MATLAB: A Tool for Engineering Analysis MATLAB Commands and Related Reference Materials Example Problem on Financial Analysis Computer Code and Results Computer Output Discussion of the MATLAB Code Code for Financial Problem 9 2 Elementary Aspects of MATLAB Graphics Introduction Overview of Graphics Example Comparing Polynomial and Spline Interpolation Conformal Mapping Example Nonlinear Motion of a Damped Pendulum A Linear Vibration Model Example of Waves in an Elastic String Properties of Curves and Surfaces Curve Properties Surface Properties Program Output and Code 65 3 Summary of Concepts from Linear Algebra Introduction Vectors, Norms, Linear Independence, and Rank Systems of Linear Equations, Consistency, and Least Squares Approximation Applications of Least Squares Approximation A Membrane Deflection Problem Mixed Boundary Value Problem for a Function Harmonic Inside a Circular Disk Using Rational Functions to Conformally Map a Circular Disk onto a Square Eigenvalue Problems Statement of the Problem Application to Solution of Matrix Differential Equations.. 94

3 x CONTENTS The Structural Dynamics Equation Computing Natural Frequencies for a Rectangular Membrane Column Space, Null Space, Orthonormal Bases, and SVD Computation Time to Run a MATLAB Program Methods for Interpolation and Numerical Differentiation Concepts of Interpolation Interpolation, Differentiation, and Integration by Cubic Splines Computing the Length and Area Bounded by a Curve Example: Length and Enclosed Area for a Spline Curve Generalizing the Intrinsic Spline Function in MATLAB Example: A Spline Curve with Several Parts and Corners Numerical Differentiation Using Finite Differences Example: Program to Derive Difference Formulas Gauss Integration with Geometric Property Applications Fundamental Concepts and Intrinsic Integration Tools in MATLAB Concepts of Gauss Integration Comparing Results from Gauss Integration and Function QUADL Geometrical Properties of Areas and Volumes Area Property Program Program Analyzing Volumes of Revolution Computing Solid Properties Using Triangular Surface Elements and Using Symbolic Math Numerical and Symbolic Results for the Example Geometrical Properties of a Polyhedron Evaluating Integrals Having Square Root Type Singularities Program Listing Gauss Integration of a Multiple Integral Example: Evaluating a Multiple Integral Fourier Series and the Fast Fourier Transform Definitions and Computation of Fourier Coefficients Trigonometric Interpolation and the Fast Fourier Transform Some Applications Using the FFT to Compute Integer Order Bessel Functions Dynamic Response of a Mass on an Oscillating Foundation General Program to Plot Fourier Expansions Dynamic Response of Linear Second Order Systems Solving the Structural Dynamics Equations for Periodic Forces Application to Oscillations of a Vertically Suspended Cable Direct Integration Methods Example on Cable Response by Direct Integration 247

4 CONTENTS xi 8 Integration of Nonlinear Initial Value Problems General Concepts on Numerical Integration of Nonlinear Matrix Differential Equations Runge-Kutta Methods and the ODE45 Integrator Provided in MAT- LAB Step-size Limits Necessary to Maintain Numerical Stability : Discussion of Procedures to Maintain Accuracy by Varying Integration Step-size Example on Forced Oscillations of an Inverted Pendulum Dynamics of a Spinning Top Motion of a Projectile Example on Dynamics of a Chain with Specified End Motion Dynamics of an Elastic Chain Boundary Value Problems for Partial Differential Equations Several Important Partial Differential Equations Solving the Laplace Equation inside a Rectangular Region The Vibrating String Force Moving on an Elastic String Computer Analysis Waves in Rectangular or Circular Membranes Computer Formulation Input Data for Program membwave Wave Propagation in a Beam with an Impact Moment Applied to One End Forced Vibration of a Pile Embedded in an Elastic Medium Transient Heat Conduction in a One-Dimensional Slab Transient Heat Conduction in a Circular Cylinder with Spatially Varying Boundary Temperature Problem Formulation Computer Formulation Torsional Stresses in a Beam of Rectangular Cross Section Eigenvalue Problems and Applications Introduction Approximation Accuracy in a Simple Eigenvalue Problem Stress Transformation and Principal Coordinates Principal Stress Program Principal Axes of the Inertia Tensor Vibration of Truss Structures Truss Vibration Program Buckling of Axially Loaded Columns Example for a Linearly Tapered Circular Cross Section Numerical Results 424

5 xii CONTENTS 10.6 Accuracy Comparison for Euler Beam Natural Frequencies by Finite Element and Finite Difference Methods Mathematical Formulation Discussion of the Code Numerical Results Vibration Modes of an Elliptic Membrane Analytical Formulation Computer Formulation Bending Analysis of Beams of General Cross Section Introduction Analytical Formulation Program to Analyze Beams of General Cross Section Program Output and Code Applications of Analytic Functions Properties of Analytic Functions Definition of Analyticity Series Expansions Integral Properties Cauchy Integral Formula Residue Theorem Physical Problems Leading to Analytic Functions Steady-State Heat Conduction Incompressible Inviscid Fluid Flow Torsion and Flexure of Elastic Beams Plane Elastostatics Electric Field Intensity Branch Points and Multivalued Behavior Conformal Mapping and Harmonic Functions Mapping onto the Exterior or the Interior of an Ellipse Program Output and Code Linear Fractional Transformations Program Output and Code Schwarz-Christoffel Mapping onto a Square Program Output and Code Determining Harmonic Functions in a Circular Disk Numerical Results Program Output and Code Inviscid Fluid Flow around an Elliptic Cylinder Program Output and Code Torsional Stresses in a Beam Mapped onto a Unit Disk Program Output and Code Stress Analysis by the Kolosov-Muskhelishvili Method Program Output and Code 550

6 CONTENTS xiii Stressed Plate with an Elliptic Hole Program Output and Code Nonlinear Optimization Applications Basic Concepts Initial Angle for a Projectile Fitting Nonlinear Equations to Data Nonlinear Deflections of a Cable Quickest Time Descent Curve (the Brachistochrone) Determining the Closest Points on Two Surfaces Discussion of the Computer Code 591 A List of MATLAB Routines with Descriptions 607 B Selected Utility and Application Functions 617 References 655 Index 663

Data-Driven Modeling. Scientific Computation J. NATHAN KUTZ OXPORD. Methods for Complex Systems & Big Data

Data-Driven Modeling. Scientific Computation J. NATHAN KUTZ OXPORD. Methods for Complex Systems & Big Data Data-Driven Modeling & Scientific Computation Methods for Complex Systems & Big Data J. NATHAN KUTZ Department ofapplied Mathematics University of Washington OXPORD UNIVERSITY PRESS Contents Prolegomenon